What is a monad-transformer as distinct from a monad?

Question

The question says it all, really. I know a (Scala) Monad looks like this:

trait Monad[M[_]] {   def pure[A](a : A) : M[A]   def bind[A, B](ma : M[A], f : A => M[B]) : M[B] } 

What does a Monad Transformer look like? And what are they used for?

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EDIT. Consider the following REPL session: if a monad transformer somehow decorates a monad with reader capabilities (or vice versa)

Let's say I just want to use replicateM from Scalaz;

scala> import scalaz._; import Scalaz._ import scalaz._ import Scalaz._  scala> some(4).replicateM[List](2) res20: Option[List[Int]] = Some(List(4, 4)) 

Now let's say, rather than having an Option[Int], I need to read an Int value from a File:

scala> val f = (_ : java.io.File) => some(1) f: (java.io.File) => Option[Int] = <function1> 

So, I can treat this reader, as if it was a Monad?

scala> ReaderT(f).replicateM[List](2) <console>:16: error: value replicateM is not a member of scalaz.ReaderT[Option,java.io.File,Int]        ReaderT(f).replicateM[List](2)                   ^ 

Err, no.

Apologies if this all seems stupid, I'm just trying to understand what lovely goodness my wrapping the File => Option[Int] in a ReaderT can actually buy me.

Answer 1

Monad transformers are type functions that when applied to a monad type, generate a new monad, that combines the behavior of both components.

E.g. in the xmonad window manager, the computations run inside:

newtype X a = X (ReaderT XConf (StateT XState IO) a) 

that is, a Reader composed with a State and an IO monad.

• Reader gives access to read-only memory
• State provides a form of read-write state
• IO allows arbitrary external effects

Note that monad transforms are thus higher-rank types. They take a monadic type of kind (* -> *), and yield a new type of that kind.

As always, the Haskell wiki has some useful content:

• Monad transformers
• Monad transformers step by step: a tutorial.

Where it all began:

• Functional Programming with Overloading and Higher-Order Polymorphism

Answer 2

Monad Transformers are used for combining / extending monads (add capabilities of one monad to another). E.g., ReaderT (Reader Transformer) enriches given monad M with Reader capabilities (transforms given monad into Reader retaining original features of M).

At the same time, Monad Transformers are normal monads that have bind, return, and other operations.

You can find examples of monad transformers in Scalaz - e.g., for Reader and State monads.